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CONTRIBUTION OF JAINA MATHEMATICIANS

CONTRIBUTION OF JAINA MATHEMATICIANS. DR. (MRS). PADMAVATHAMMA, M Sc, Ph D Professor of Mathematics (retired) Department of Studies in Mathematics University of Mysore, Manasagangotri Mysore-570 006 E-mail: padma_vathamma@yahoo.com.

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CONTRIBUTION OF JAINA MATHEMATICIANS

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  1. CONTRIBUTION OF JAINA MATHEMATICIANS DR. (MRS). PADMAVATHAMMA, M Sc, Ph DProfessor of Mathematics (retired)Department of Studies in MathematicsUniversity of Mysore, ManasagangotriMysore-570 006E-mail: padma_vathamma@yahoo.com

  2. Dr.Padmavathamma

  3. Mathematics is one of the important branches of Science from time immemorial. Being an inseparable part of science, it has retained its privileged place as Queen of all Sciences. • The contribution of Indian mathematicians towards the development of mathematics is unique and valuable. • Zero was first introduced in place value system of notations by Indians. • The contributions of ancient Indian mathematicians Āryabhaṭa, Bhāskara, Brahmagupta, Mahāvīrācārya and Bhāskarācārya are world famous even today. • Many Jaina mathematicians have saliently contributed. It is perceived among common people that mathematics is difficult to learn. The skill of explaining such difficult material in simple and exact forms is one of the specialties of Jaina mathematicians. Dr.Padmavathamma

  4. Tattvārthadhigama Sūtra, Sthānanga Sūtra, Jambūdvīpa Prajñapti, Tiloyapaṇṇatti, Kṣetrasamāsa, Gaṇitasārasangraha and Vyavahāragaṇita are important ancient Jaina mathematical works. • Among the important subjects which are available in Jainā philosophy, first priority is given to literature while the second priority is given to mathematics. Hence in Āgamās, it is said lehāiyāvo gaṇiyappahāṇāo, that is, the writing etc. of which the chief (pradhāna) is the counting. From this, it is proved that in educating a child and in the human transactions, mathematics had a very prominent role. Dr.Padmavathamma

  5. In Jainā literature there are four anuyogās called prathama, karaṇa, caraṇa and dravya. In karaṇānuyoga, many mathematical operations are used to explain the features of loka and in the explanations of sun, moon, star, island, sea etc.we find the use of mathematics in the following Prakrit works and their commentaries. • Sūryaprajñapti, Candraprajñapti, Jambūdvīpaprajñapti, Tiloyapaṇṇatti, Dhavalā commentaries of Ṣaṭkhaṇḍāgama, Gommaṭasāra, Trilokasāra. • The above works provide valuable information to know the ancient Indian mathematics. Sūryaprajñapti is called Gaṇitānuyoga. Dr.Padmavathamma

  6. The following list shows the names of persons who have contributed to the development of mathematics in Prakrit. • Puṣpadanta- Bhūtabali (3rd century A.D) • Yativṣabhācārya (5th century A.D) • Vīrasenācārya (9th century A.D) • Srīdhara (9th century A.D) • Srīpati (10th century A.D) • NemicandraSiddhāntaCakravarti (11th century A.D) • We do find names of Siddhasena, Bhadrabāhu etc. who have used mathematical formulae in their works, although they were not mathematicians. Dr.Padmavathamma

  7. For a Jaina monk, arithmetic and astrology were like adornments. Umāsvāti (150 B.C) who was one of the best Jainā philosophers, had mentioned for the first time about the mathematics school at Kusumaoura in Pāṭna. He was living in the ancient Pāṭalīpura which is now the modern Kusumapura in Paṭna. • It is quite possible that this mathematics school existed even before the time of the famous Jainā monk Bhadrabāhu (300 B.C) who lived in Kusumapura). His works are – a commentary on Sūryaprajñapti, and Bhadrabāhavi Samhita. Dr.Padmavathamma

  8. Ancient India has contributed a lot to the development of mathematics and the part played by the Jainā scholars in this field is significant. The development of mathematics in India may be classified into the following groups. • Initial period (Ādikāla) - 3000 – 500 B.C • Childhood Period (Śaiśavakāla) – 500 B.C -500 A.D • This is also known as Dark Period • Middle Period – 500 – 1200 A.D • Later Period – 1200 -1800 A.D • Modern Period – 1800 A.D onwards Dr.Padmavathamma

  9. Mathematical quotations in Ardhamāgadhi and Prakrit are met with in several works. Dhavalā contains a large number of such quotations. A.N.Singh in his article entitled, Mathematics of Dhavalā says “ A study of the Jainā canonical works reveals that mathematics was held in high esteem by the Jainās. In fact, the knowledge of mathematics and astronomy was considered to be one of the principal accomplishments of the Jainā ascetics. ” Dr.Padmavathamma

  10. Mathematical material in the Dhavalā may be taken to belong to the period 200 – 600 A.D. Thus Dhavalā becomes a work of first rate importance to the historians of Indian mathematics – the period preceding the fifth century A.D called the dark period. • In the present paper we discuss about the following three authors • NemicandraSiddhāntaCakravarti whose works are in Prakrit • Mahāvīrācārya whose works are in Sanskrit • Rājāditya whose works are in Kannada Dr.Padmavathamma

  11. NemicandraSiddhāntaCakravartiINTRODUCTION • There are four celebrated ascetic sanghās in the History of Jainās in South India. These sanghās are Nandi, Simha, Sena and Deva. • The Deśīyagaṇa is a branch of the Nandi sangha and originated in the lands called Deśa which extended from river Cauvery in the south to river Godāvari in the north, Sahyadri hills in the east to Palghat in the west (present day Kerala). • Jain ācāryaSimhanandi belonging to this gaṇa helped Sivamara to found the Ganga dynasty, one of the ancient royal kingdoms of India [1]. Dr.Padmavathamma

  12. Of the twenty scholars in this gaṇa who were honoured with the title “SiddhāntaCakravarti” , ācāryaNemicandra is most known for his work in mathematics. • Ascetic Lineage of Nemicandra: It is known from the Gommaṭasāra (Karmakāṇḍa part) that Nemicandra was the disciple of Abhayanandi, whose preceptor was Guṇanandi. Vīranandi, a colleague and contemporary of Nemicandra was alsoa disciple of Abhayanandi. Vīranandi, the author of the CandraprabhaMahākāvya, has received homage from Nemicandra in the Gommaṭasāra (Chapter 6, verse 396). Dr.Padmavathamma

  13. Nemicandra as a mathematician • Except the Trilokasāra which gives cosmological description, all other works of Nemicandra are related to Jain philosophy. • His profound knowledge of mathematics i.e rules related to circle and its segments, permutations and combinations are all employed in his works • However, the earlier mathematicians in India had also known this science, before him. It is known, however, some of his examples and illustrations on combinations were never seen in the Hindu mathematics. • The pioneering research work of Prof.B.Datta [2] and Prof.L.C.Jain [3] would definitely throw more light on the mathematics of Nemicandra. Dr.Padmavathamma

  14. The works of Nemicandra • NemicandraSiddhāntaCakravarti is the author of Dravyasamgraha, Gommaṭasāra, Labdhisāra, Kśapaṇasāra and Trilokasāra. This paper is mostly concerned with the first two and the fourth works to explore their mathematical approach. • Art present, Prof.L.C.Jain has studied the mathematical and scientific matters contained in the Labdhisāra under the auspicious of Indian National Science Academy. • He has worked out mathematical and system theoretical aspects including explanations of algebraic and geometric expressions, based on the verses of the commentaries. Dr.Padmavathamma

  15. In the study, he has found numerical symbols (anka samdśṭi) in both the Labdhisāra and in its commentaries [4]. Furthermore, he has compiled a comprehensive glossary of technical terms relevant to Labdhisāra. Dr.Padmavathamma

  16. THE MATHEMATICS OF NEMICANDRĀ’S WORKS • Before stating the laws of indices of Nemicandra it is better to give his terminology. If N = 2n tyhen n is called the ardhacheda of N. • How many times a given number can be halved will be the ardhacheda. Sometimes the word ardha is left out and only cheda is used. In general, if N = xn then n is the cheda of N with respect to the base x. If N = 22n then n is called the ardhacheda of the ardhacheda of N Dr.Padmavathamma

  17. Nemicandra gives the following rule: If the ardhacheda of the multiplication is added to the ardhacheda of the multiplier, then the ardhacheda of the product is obtained. This will mean more chedas as shown in the following formula: 2m x 2n = 2m+n • If the ardhacheda of the divisor is subtracted from the ardhacheda of the dividend then the ardhacheda of the quotient is obtained. 2m ÷ 2n = 2m-n Dr.Padmavathamma

  18. If the distributed number is multiplied by the substituted number then the ardhacheda of the resulting number is got. • This means that if m is distributed into its units and each similar unit is replaced by N then the resulting number is R = Nm . If N = 2n then according to the rule we get the following R = 2nm . • If the ardhacheda of the distributed number is added to the ardhacheda of the substituted number then vargaśalāka of the resulting number is obtained. From the rule of Nemicandra it is evident that he knew the following rules of indices: xmx xn = xm+n , xm ÷ xn = xm-n , (xm)n = xmn • In Trilokasāra fourteen types of series are used to explain the samkhyamāna and upamamāna. Dr.Padmavathamma

  19. Arithmetical Progressions The following rule is given by Nemicandra in relation to arithmetical progressions. • Multiply the number obtained by subtracting the number of terms by one and the common difference. If the product is added to the first term then the last term is obtained and if this product is subtracted from the last term then the first term is obtained. If half the sum of the first and the last terms are multiplied by the number of terms, then the sum of the series is obtained. Dr.Padmavathamma

  20. Geometrical Progressions To find out the sum of a geometrical progression Nemicandra provides the following rule: • Multiply the common ratio as many times as the number of terms. Subtract one from the product and then divide by the number obtained by subtracting one from the common ratio and multiply by the first term. The resulting number will be the sum of the geometrical progression. • Algebraically this can be written as S = a (rn - 1) ÷ (r – 1), where a is the first term, r is the common ratio and S is the sum of the series. Dr.Padmavathamma

  21. Circle Nemicandrā’s rule regarding circles is included in the following: • The (gross) circumference of a circle will be three times its diameter. • The (accurate) circumference is the square-root of ten times the square of the diameter. • The accurate area is obtained if we multiply one-fourth of the diameter and the circumference. Here the value of π is taken as √10. • To determine the accurate circumference and area of the Jambūdvīpa the second law is used. Dr.Padmavathamma

  22. The Prism, Cone and Sphere • According to Nemicandra the volume of the prism = (area of ) base x height, the volume of the cone = (1\3) base x height and the volume of the sphere = (9\2) (radius)3 • For example, to measure the volume of a heap of (mustard like) seeds which resemble a cone, Nemicandra has given the following formula. Volume = (circumference\6)2 x height • In such cases it was supposed that height = (1\11) circumference. And finally we may simply note at this juncture that Nemicandra has also provided mathematical rules regarding the segments of a circle, a trapezium and many other permutations and combinations. Dr.Padmavathamma

  23. REFERENCES • S.C.Ghośal,Dravyasangraha, MotilalBanarsidas, New Delhi, 1989 • B.Datta, Mathematics of Nemicandra, Jainā Antiquary, Vol.1,No.2, Arrah, 1935, p.25-44 • L.C.Jain, Divergent Sequences Locating Transfinite Sets in Trilokasāra, Indian Journal of History of Science, Vol.12, No.1, 1977, p.57-75 • The Labdhisāra, Vol.1, mSSMK Jain Trust, Katni,India, 1994, p.5 Dr.Padmavathamma

  24. Introduction • Mahāvīrācārya was a famous Jaina mathematician who succeeded after the well-known scholars Āryabhaṭa (C. 5th century A.D.), Varāhamihira (C. 6th century A.D.) and Brahmagupta (C. 7th century A.D.). Not much is known about his life. • According to the literature available, he hailed from Karnataka. • He enjoyed the patronage of the Rāṣtrakūṭa king Amoghavarṣa Npatuṅga, who ruled in Mānyakheṭa (South India) from 815 A.D. to 877 A.D. • The period of Npatuṅgā's rule was well-known for political stability and the development of art and culture. Dr.Padmavathamma

  25. Mahāvīrācārya is the author of the Sanskrit work Gaṇitasārasaṅgraha (abbr. GSS), which is on elementary mathematics. • It is a compilation work on universal (Laukika) mathematics which is based on non-universal (Lokottara) mathematics contained in Jaina Āgamās. • It provides a valuable source of information on ancient Indian mathematics. The Gaṇitasārasaṅgraha was not available for a long time. Dr.Padmavathamma

  26. Discovery of Gaṇitasārasaṅgraha • Professor M. Rangācārya was appointed as a professor of Sanskrit and comparative philology at the Presidency College, Madras (Chennai). • He also took charge of the office of the Government Oriental Manuscripts Library. The Director of Public Instruction, Mr. G. H. Stuart, directed Rangācārya to find out whether there were any manuscripts in the library which could throw new light on the History of Hindu Mathematics. • If so, to publish it with an English translation and notes. Rangācārya first found three incomplete manuscripts of Mahāvīrācaryā's GSS. Dr.Padmavathamma

  27. Out of the three manuscripts, one is written on paper in Grantha characters with a running commentary in Sanskrit. • The other two are palm-leaf manuscripts in Kanarese characters, which contain a brief statement in the Kanarese language of the figures, relating to the various illustrative problems as also of the answers to the same problems. • At the instance of Mr. G.H. Stuart, Prof. Rangācārya tried to get more manuscripts from other places and finally succeeded in getting two more manuscripts. Dr.Padmavathamma

  28. One was from the Government Oriental Library at Mysore. • This is a transcription on paper in Kanarese characters of an original palm-leaf manuscript. It contains the whole of the work with a short commentary in Kannada by Vallabha. • The other, that is, the fifth manuscript is also a transcription on paper in Kanarese characters of a palm-leaf manuscript found in the Jain Math at Muḍbidri (South Canara). • This manuscript also contains the whole work and gives a brief statement of the problems and their answers. Dr.Padmavathamma

  29. After carefully studying and examining these five manuscripts, Prof. Rangācārya was successful in translating Gaṇitasārasaṅgraha into English and in writing mathematical notes wherever necessary. • The Madras Government published this valuable work in 1912 [18]. • Dr. D.E. Smith, Professor of Mathematics, Teacher's College, Columbia University, New York, has written an introduction to this book. In fact, he had read a paper on GSS at the fourth International congress of Mathematicians held at Rome in April 1908. Dr.Padmavathamma

  30. Translators / Commentators of GSS • Vallabha (Daivajña-Vallabha) has written commentaries both in Kannada and Telugu for GSS. • Pāvalurimallaṇṇa [2] has translated GSS into Telugu. • From D. Pingree [17] it is clear that a commentary by Varadarāja and a Rājasthāni translation by Amicandra (in 1842) are also there. L.C. Jain translated GSS into Hindi in 1963 and edited it with collation of additional manuscripts and with detailed historical introduction from beginning of the histrical era upto Mahāvīrācārya. • This was published by Jain Samskriti Samrakṣka Sangha, Sholapur [9]. Dr.Padmavathamma

  31. New Edition of Gaṇitasārasaṅgraha • Mahāvīrācārya happens to be from Karnataka, the motherland of Kannada. • There was a need for the Kannada version so that Kannadigas could appreciate the beauty of the mathematics contained in GSS. • This deficiency was met by the author. • She has translated the original Sanskrit verses into Kannada and also translated the Sanskrit verses into English. • Since both the English and Hindi editions of GSS were out of print for a long time, the English translation was also included in the new work. • This new edition of GSS [16] was published by Sri SiddhāntaKeerthi, Granthamala of Sri Hombuja Jain Math, Shimoga District, Karnataka in the year 2000. Dr.Padmavathamma

  32. This new edition of GSS is a rare and unique text. It appears as a combination of three fine fragrant flowers blooming from the same creeper. • The new style and the format followed in this book are of high standard and very attractive. • The Sanskrit, English and Kannada versions which are like the gemtrios are accommodated in the same volume. • Similar to the mingling of three sacred rivers, this single book embodies the presentation of the text in three different languages- Sanskrit, English and Kannada. • The review of this book by S. Balachandra Rao has appeared in Gaṇitabhārati, vol. 25, Nos. 1-4, 2003, p. 197-199. Dr.Padmavathamma

  33. TELUGU EDITION OF GSS • GSS was translated into Telugu by Vidwan T. Subbarao and edited by P.V. Aruṇāchalam. This was published [21] by the Telugu Academy, Hyderabad in the year 2003. Dr.Padmavathamma

  34. Other works of Mahāvīrācārya • Different research scholars have agreed that Mahāvīrācārya is also the author of the following four works: 1. Ṣattrimśika (Ṣattrim Ṣatika) 2. Jyotiṣapaṭala 3. Kṣetra Gaṇita 4. Chattīsapūrva Uttara Pratisaha Dr.Padmavathamma

  35. Mathematics of GSS • The style of GSS is in the form of a text book. • This is a collection of the South Indian mathematics which was embedded in DigambaraJaina texts of the Karaṇānuyoga and the Dravyānuyoga Groups. • Keeping in view the Jaina Karma Theory in its Pūrvā's tradition of the DigambaraJaina School, it can be said that the mathematics of GSS is the one that has come through mathematico philosophical texts of the DigambaraJainācāryās and definitely not the sole contribution of Mahāvīrācārya alone. • This is made clear by Mahāvīrācārya himself in the following stanzas 17, 18, and 19 of the first chapter of GSS. However the rules and formulae in the existing literature which appear first in GSS can be certainly credited to him: Dr.Padmavathamma

  36. Translation – Sanskrit shloka • With the help of the holy sages, who are worthy to be worshipped by the lords of the world and of their disciples and disciples, who constitute the well-known jointed series of preceptors, I glean from the great ocean of the knowledge of numbers a little of its essence, in the manner in which gems are (picked up) from the sea, gold is from the stony rock and the pearl from the oyster shell and give out, according to the power of my intelligence, the Sārasaṅgraha, a small work on arithmetic, which is (however) not small in value. Dr.Padmavathamma

  37. The List of Chapters Detailed in GSS 1.Samjñadhikārah(On Terminology) 2.Parikarma Vyavahārah (Treatment on Algebraic operations) 3. Kalā Savarṇa Vyavahārah (Treatment on Fractions) 4.Prakīrṇaka Vyavahārah (Miscellaneous Problems on Fractions) 5. Trairāśika Vyavahārah (The Rule of Three) 6.Miśraka Vyavahārah (Mixed Problems) 7.Kṣetragaṇita Vyavahārah (Calculations Relating to the Measurement of Areas) 8.Khāta Vyavahārah(Calculations Regarding Excavations) 9. Chāyā Vyavahārah (Calculations Relating to Shadows) Dr.Padmavathamma

  38. Chapters I and II deal with the six algebraic operations multiplications, division, squaring, cubing, extraction of square roots and cube roots. Arithmetic and geometric series have also been discussed. • In case of multiplication, four rules are given with examples. Along with the rule of division, the modern rule is also explained. There is a special rule for squaring which is as follows: SANSKRIT SHLOKA Dr.Padmavathamma

  39. Translation – Sanskrit shloka • Get the square of the last figure (in the number, the order of counting the figures being from the right to the left) and then multiply this last (figure), after it is doubled and pushed on (to the right by one notational place), by (the figures found in) the remaining places. Each of the remaining figures (in the number) is to be pushed on (by one place ) and then dealt with similarly. This is the method of squaring. Dr.Padmavathamma

  40. This rule will be clear from the following examples: 1. To find the square of 12 12 = 1 2 × 1 × 2 = 4 22 = 4 Therefore the square of 12 = 144 2. To find the square of 131 12 = 1 2 × 1 × 3 = 6 2 × 1 × 1 = 2 32 = 9 2 × 3 × 1 = 6 12 = 1 Therefore the square of 131 = 17 1 6 1 Dr.Padmavathamma

  41. Mahāvīrācārya has discussed various algebraic operations involving zero. He writes (GSS, chapter 1, Verse No. 49) : Dr.Padmavathamma

  42. Translation – Sanskrit shloka • A number multiplied by zero is zero and that (number) remains unchanged when it is divided by ; combined with or diminished by zero. • Multiplication and other operations in relation to zero (give rise to) zero and then in the operation of addition, the zero becomes the same as what is added to it. • Algebraically, the above can be expressed as follows [16, p.10]: A × 0 = 0, A + 0 = A, A – 0 = A, A 0 = A • Actually Mahāvīrācāryā's rule related to multiplication, addition and subtraction is correct, but his rule related to division has another interpretation, implying that the division by zero means the non-existence of divisor. Sridhara (c. 10th century A.D) who was, perhaps, not earlier to Mahāvīrācārya has not considered division by zero. Dr.Padmavathamma

  43. It is a point to be noted that according to concept of improper or mathematical infinity, the correct answer as a limiting value was known to Brahmagupta 300 years earlier. • Bhāskarācārya (1150 A.D.) has given the symbol Khaharafor the result of division by zero and rightly assigns to it the value of mathematical infinity. • The concept of proper infinities in the Jaina Āgamās was to come with George Cantor in the sixties of the nineteenth century, Mahāvīrācārya obviously thinks that a division by zero is not division at all. • Multiplications of those numbers which lead to numbers of a necklace (that is numbers which are same when read either from right or left) are very interesting : [16, Chapter II, Examples 3 et seq.] Dr.Padmavathamma

  44. 139 × 109 = 15151 152207 × 73 = 11111111 14287143 × 7 = 100010001 12345679 × 9 = 111111111 142857143 × 7 = 1000000001 11011011 × 91 = 1002002001 27994681 × 411 = 12345654321 333333666667 × 33 = 11000011000011 • Chapters III and IV deal respectively with kalāsavarṇa and Prakīrṇaka Vyavahāra and are completely devoted to fractions. Types of fractions and operations on fractions have been discussed in detail. Some points worthy to be noted are given below. Dr.Padmavathamma

  45. Credit goes to Mahāvīrācārya for expressions of unit fractions as the sum of unit fractions. In words of Brijmohan [5], "No other Indian mathematician has even touched upon this". This problem has created much interest to Ahmes (Papyrus, 1050 B.C.). In modern notation the relevant problems [16, Chapter III, 75-78] can be expressed as follows: Dr.Padmavathamma

  46. Dr.Padmavathamma

  47. In Chapter V, Mahāvīrācārya has given utmost importance to Rule of Three and most part of it is devoted to Rule of Three and its generalised forms. • In JainaĀgamās, permutations and combinations play an important role. • As a result in Gaṇitasārasaṅgraha also it has been explained in Chapter VI in great detail. • Actually the following rule gives the number which can be chosen (out of n given things) r at a time. The concerned verse is [16, Chapter VI, verse No. 218] :- Dr.Padmavathamma

  48. Translation – Sanskrit shloka • Beginning with one and increasing by one, let the numbers going up to the given number of things be written down in regular order and in the inverse order (respectively) in an upper and a lower (horizontal) row, (If) the product (of one, two, three or more of the numbers in the upper row) taken from right to left be divided by the (corresponding) product (of one, two, three or more of the numbers in the lower row) also taken from right to left, (the quantity required in each such case of combination) is (obtained as) the result. Dr.Padmavathamma

  49. Algebraically, • The credit goes to Mahāvīrācārya for the above formula since he appears to be the first to collect it in the world as above. It is also interesting to note that the same formula became prevalent again through Herign in 17th century A.D. Dr.Padmavathamma

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